Enhancement of charged macromolecule capture by nanopores in a salt gradient

Abstract

Nanopores spanning synthetic membranes have been used as key components in proof-of-principle nanofluidic applications, particularly those involving manipulation of biomolecules or sequencing of DNA. The only practical way of manipulating charged macromolecules near nanopores is through a voltage difference applied across the nanopore-spanning membrane. However, recent experiments have shown that salt concentration gradients applied across nanopores can also dramatically enhance charged particle capture from a low concentration reservoir of charged molecules at one end of the nanopore. This puzzling effect has hitherto eluded a physically consistent theoretical explanation. Here, we propose an electrokinetic mechanism of this enhanced capture that relies on the electrostatic potential near the pore mouth. For long pores with diameter much greater than the local screening length, we obtain accurate analytic expressions showing how salt gradients control the local conductivity which can lead to increased local electrostatic potentials and charged analyte capture rates. We also find that the attractive electrostatic potential may be balanced by an outward, repulsive electro-osmotic flow that can in certain cases conspire with the salt gradient to further enhance the analyte capture rate.

Figure 1

Schematic of electrokinetic focusing experiments. (a) Charged analytes are placed in the right reservoir and a voltage bias is applied. (b) The structure of the pore, electrostatic potential, and electro-osmotically driven fluid flow in dimensionless units, with distance measured in units of the pore radius. For long pores (a∕d=w⪡1), the concentration and potential fields are approximated as constant within the small hemispherical regions capping the pore (denoted by ±). Field and flux continuity conditions are applied at the hemispherical surfaces. The electro-osmotic flow field in the right chamber will be approximately spherically symmetric (red) if the membrane surface is uniformly charged but will be more lobelike (black) if the membrane flange is uncharged and no-slip boundary conditions are imposed (Refs. 1, 2).

Figure 2

Salt concentration and electrostatic potential across an unimpeded pore of aspect ratio w=a∕d=0.1. (a) The salt density Σ(ξ) as a function of the axial coordinate for various EOF velocities U and salt ratio Σ_L=3. (b) The normalized potential Φ(ξ)∕V for various salt ratios Σ_L. The flat segments in both plots correspond the hemispherical cap regions in which all quantities are approximated as constant. The errors introduced in the quantities outside the caps with such an approximation are of order w².

Figure 3

(a) EOF velocity U as a function of effective pore EOF permeability Γ_R. The response deviates from linear for large and small salt ratios. The deviations are most pronounced for Γ_R>0 where EOF is into the right reservoir (U>0). (b) EOF velocity as a function of salt ratio for various effective pore surface charge densities. For Γ_R>0, increasing the salt ratio decreases the effective screening length in the pore, reducing the EOF velocity [cf. Eq. 26]. When Γ_R<0, the salt in the right reservoir is swept into the left reservoir, keeping the screening length approximately ${\kappa }_R^{-1}$ throughout the pore and very little dependence on Σ_L arises. In both plots V=10 and w=a∕d=0.2.

Figure 4

(a) The electrostatic potential in the right reservoir (normalized by the applied potential V), as a function of salt asymmetry Σ_L for various pore EOF permeabilities Γ_R. Here the pore aspect ratio was set to w=a∕d=0.2. (b) The magnitude of the potential near the mouth pore as a function of EOF permeability Γ_R for various salt ratios.

Figure 5

(a) Pore mouth occupation fraction θ as a function of salt ratio Σ_L for various EOF permeability Γ_R and V=10. Note the sharp increase in θ as a function of Σ_L for positive surface charges. Parameters used were ρ_∞=10⁻⁷, q=30, V=10, w=0.2, k_on∕k_off=1000, and k_on∕k_t=10⁴. For small Σ_L, larger Γ_R>0 induces larger U>0, pushing the analyte away. At higher salt Σ_L, the EOF is mitigated due to the reduction in ζ-potential (or effective surface charge) indicated by Eq. 27. The reduction of EOF to modest values allows the attraction from the term qΦ₊ to overcome the repulsive effect of the EOF [cf. Fig. 4a], increasing θ. (b) Occupation fraction θ as a function of bias voltage V at different salt ratios and Γ_R=0.1.

Figure 6

(a) Normalized capture rate Ω_c as a function of bias voltage. (a) The effect of varying salt ratio Σ_L. (b) The effect of varying k_on∕k_t=ω_on∕ω_t with fixed Σ_L=2.